By doppler shift I assume you mean red-shift and blue-shift? They would affect the energy imparted by photons with respect to the photoelectric effect. The photoelectric effect doesn’t know that light was originally blue in color, if it arrives as red photons. Red shifted light would have less energy. Same with blue-shifted light having more energy. Kind of how if someone throws a projectile at you, it will impart a certain amount of energy onto you, but if you’re moving towards the projectile, the amount of energy imparted will increase since the relative kinetic energy increases.
So E=hf would still hold, h is constant so as the frequency of incident light changes, the energy would also change.
I believe that Amrit's answer is wrong. With the photolelectric device travelling at sufficient speed away from the source of light, any frequency could be red-shifted below the photoelectric threshold, as Osamah says. Conversely, travelling towards the source of light at sufficient speed would increase a frequency from below the threshold to above it, making the difference between no electricity and some electricity.
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